Class 7 Mathematics Ch 5 Mastery Guide: Set Theory, Subsets, Venn Diagrams, Union, Intersection, Difference & De Morgan Laws (FBISE)
Instructional Guide: Unit 05 Sets
- Define a set as a well-defined, distinct collection and use membership symbols ($\in, \notin$).
- Represent sets in descriptive and tabular (roster) forms; understand standard number sets ($\mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{P}, \mathbb{E}, \mathbb{O}$).
- Define finite, infinite, empty (null $\emptyset$ or $\{\}$), and universal ($\mathcal{E}$) sets.
- Differentiate between subsets ($\subseteq$) and supersets ($\supseteq$).
- Differentiate between proper subsets ($\subset$) and improper subsets (the set itself).
- Compute the total number of subsets ($2^n$) and proper subsets ($2^n - 1$) for a set of cardinality $n$.
- Distinguish between equal sets ($A = B$) and equivalent sets ($A \leftrightarrow B$).
- Identify disjoint sets ($A \cap B = \emptyset$) and overlapping sets ($A \cap B \neq \emptyset$).
- Perform fundamental set operations: Union ($A \cup B$), Intersection ($A \cap B$), and Difference ($A - B$ and $B - A$).
- Find the Complement of a set ($A' = \mathcal{E} - A$) and verify complement properties.
- Verify De Morgan's Laws: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.
- Represent all set relationships and operations pictorially using Venn diagrams.
- Class 5 & 6 Foundations: Recognizing collections of items, prime vs composite numbers, factors, and multiples.
- Number Sets: Natural numbers $\{1, 2, 3, \dots\}$, whole numbers $\{0, 1, 2, \dots\}$, and integers $\{0, \pm 1, \pm 2, \dots\}$.
- Grouping & Sorting: Sorting physical items by shared characteristics (e.g., shapes, colors, categories).
- Element vs. Subset: $x \in A$ means $x$ is an individual item inside $A$. In contrast, $\{x\} \subseteq A$ is a set containing $x$. An element is never a subset unless wrapped in set brackets!
- Proper vs. Improper Subset: An improper subset is the original set itself. Every other subset is proper. Every set with $n$ elements has exactly ONE improper subset and $2^n - 1$ proper subsets.
- Null Set Property: The empty set $\emptyset$ is a subset of EVERY set ($\emptyset \subseteq A$), but it is NOT an element of $A$ unless $A$ explicitly contains $\emptyset$ as an item.
- Equal vs. Equivalent: Equal sets have the exact same members ($A = B$). Equivalent sets merely have the same number of members ($n(A) = n(B)$). All equal sets are equivalent, but not all equivalent sets are equal!
- Difference Non-Commutativity: $A - B \neq B - A$. $A - B$ removes elements of $B$ from $A$, whereas $B - A$ removes elements of $A$ from $B$.
- De Morgan's Operator Flip: When taking the complement of a union or intersection, the operator flips: $(A \cup B)' = A' \cap B'$ (union becomes intersection), and $(A \cap B)' = A' \cup B'$ (intersection becomes union).
Introduce sets through real-world categorization: a postage stamp collection divided into themes (historical monuments, birds, national heroes), a school library organized into fiction and science, or a playlist of favorite songs. In computer science and modern databases (SQL), every search filter (e.g. "Shoes under Rs. 3000 AND size 8") is a set intersection ($A \cap B$)!
The Birth of Set Theory: Georg Cantor's Mathematical Revolution
Historical Spotlight: Georg Cantor (1845–1918)
Georg Cantor, a renowned German mathematician, is revered as the founder of Set Theory. Before Cantor, mathematics treated numbers as isolated quantities. Cantor asked a profound question: "What happens when we collect mathematical objects into a single whole?" His revolutionary work laid the bedrock of modern mathematics, theoretical computer science, and data structures.
Well-defined collection
Distinct elements
Membership: $\in, \notin$
Subsets: $B \subseteq A$
Proper: $B \subset A$
Count: $2^n$ & $2^n - 1$
Equal ($A = B$)
Equivalent ($n(A)=n(B)$)
Disjoint & Overlapping
Union: $A \cup B$
Intersection: $A \cap B$
Difference: $A - B$
Universal: $\mathcal{E}$
Complement: $A' = \mathcal{E} - A$
De Morgan's Laws
Section 1: What is a Set? Notations, Subsets & Set Relationships
Mastering fundamental set terminology: elements, cardinality, subsets, equality, and disjointness.
1.1 Definition of a Set
A set is a collection of well-defined and distinct objects.
- Well-defined: A specific rule or property enables anyone to determine conclusively whether a given object belongs to the collection or not. E.g., "The set of odd numbers less than 10" is well-defined. But "The set of delicious foods" or "The set of tall students" is NOT well-defined because tastes and heights are subjective.
- Distinct: No object is repeated in a set. Each element is written only once. In the word
"WOOD", the letters form the set $\{ ext{W, O, D}\}$.
Standard Mathematical Number Sets
Membership Symbols & Cardinality
- The Greek symbol $\in$ means "belongs to" or "is an element of" (e.g., $2 \in \{2, 3, 5\}$).
- The symbol $\notin$ means "does not belong to" (e.g., $4 \notin \{2, 3, 5\}$).
- Cardinality $n(A)$: The total count of distinct elements contained in set $A$. For $A = \{a, b, c, d, e\}$, $n(A) = 5$.
Subsets, Supersets, Proper & Improper Subsets
If every element of set $B$ is also an element of set $A$, then $B$ is a subset of $A$ ($B \subseteq A$), and $A$ is a superset of $B$ ($A \supseteq B$).
$B$ is a proper subset ($B \subset A$) if all elements of $B$ are in $A$ and $A$ contains at least one element not in $B$ ($B \neq A$).
The set itself ($A \subseteq A$) is called the improper subset of $A$.
For any set containing $n$ elements:
$$\text{Total Number of Subsets} = 2^n$$
$$\text{Total Number of Proper Subsets} = 2^n - 1$$
Example: For $S = \{a, b, c\}$ with $n = 3$: Total subsets $= 2^3 = 8$. Proper subsets $= 8 - 1 = 7$.
The 8 subsets are: $\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}$ (7 proper subsets) and $\{a, b, c\}$ (1 improper subset).
Equal vs. Equivalent & Disjoint vs. Overlapping
- Equal Sets ($A = B$): Contain the exact same elements. E.g., $\{1, 2, 3\} = \{3, 1, 2\}$.
- Equivalent Sets ($A \leftrightarrow B$): Contain the same number of elements ($n(A) = n(B)$), establishing a one-to-one correspondence. E.g., $\{a, b, c\} \leftrightarrow \{1, 2, 3\}$.
- Disjoint Sets: Share zero common elements ($A \cap B = \emptyset$). E.g., $\{1, 2\}$ and $\{3, 4\}$.
- Overlapping Sets: Share at least one common element, but neither is a subset of the other ($A \cap B \neq \emptyset$, $A \not\subseteq B$, $B \not\subseteq A$). E.g., $\{1, 2, 3\}$ and $\{2, 3, 4\}$.
Complete Step-by-Step Solutions: Exercise 5.1
12 Questions • 100% Solved
For each of the following sets, show all the elements of the set:
(i) $A$ is the set of positive integers between $5$ and $9$.
(ii) $B$ is the set of common multiples of $2$ and $3$.
(iii) $C$ is the set of factors of $66$.
Part (i): "Between 5 and 9" excludes the endpoints 5 and 9. The positive integers between 5 and 9 are 6, 7, and 8.
$$A = \{6, 7, 8\}$$
Part (ii): Common multiples of 2 and 3 are multiples of $\text{LCM}(2, 3) = 6$. These are $6, 12, 18, 24, \dots$
$$B = \{6, 12, 18, \dots\}$$
Part (iii): Factors of 66 are numbers that divide 66 completely with zero remainder: $1 \times 66, 2 \times 33, 3 \times 22, 6 \times 11$.
$$C = \{1, 2, 3, 6, 11, 22, 33, 66\}$$ Final Answers: (i) {6, 7, 8} | (ii) {6, 12, 18, ...} | (iii) {1, 2, 3, 6, 11, 22, 33, 66}
For each of the following sets, describe the set by listing all its elements:
(i) $P = \{\text{names of the days of the week}\}$
(ii) $Q = \{\text{colours of a rainbow}\}$
(iii) $R = \{\text{odd numbers between 10 and 26}\}$
(iv) $S = \{\text{factors of 15}\}$
(v) $T = \{\text{multiples of 3 between 1 and 14}\}$
Part (i): The 7 days of the week are:
$$P = \{\text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\}$$
Part (ii): The 7 colours of the rainbow (VIBGYOR) are:
$$Q = \{\text{Violet, Indigo, Blue, Green, Yellow, Orange, Red}\}$$
Part (iii): Odd numbers strictly between 10 and 26:
$$R = \{11, 13, 15, 17, 19, 21, 23, 25\}$$
Part (iv): Numbers dividing 15 completely ($1 \times 15, 3 \times 5$):
$$S = \{1, 3, 5, 15\}$$
Part (v): Multiples of 3 strictly between 1 and 14 ($3 \times 1 = 3, 3 \times 2 = 6, 3 \times 3 = 9, 3 \times 4 = 12$):
$$T = \{3, 6, 9, 12\}$$
If $A = \{1, 2, 3, 4, 5\}$, write down the value of $n(A)$.
$$n(A) = 5$$ Final Answer: n(A) = 5
Given the sets: $A = \{1, 2, 3, 4, 5, 6, 7\}$, $B = \{3, 4, 5\}$, and $C = \{4, 5, 6, 7, 8\}$. State true or false in each case:
(i) $B \subset A$
(ii) $C \subset A$
(iii) $C \subset B$
Part (i): Every element of $B$ ($3, 4, 5$) belongs to $A$, and $A$ contains other elements ($1, 2, 6, 7$). Hence $B$ is a proper subset of $A$. → True.
Part (ii): $C$ contains the element $8$, which does NOT belong to $A$ ($8 \notin A$). Hence $C$ is not a subset of $A$. → False.
Part (iii): $C$ contains elements $6, 7, 8$ which do not belong to $B$. → False.
Final Answers: (i) True | (ii) False | (iii) False
Given $A = \{2, 4, 6, 8\}$ and $B = \{\text{even numbers less than 10}\}$. Which of the following statements is true and which is false?
(i) $A \subseteq B$ (ii) $A \subset B$ (iii) $A \not\subseteq B$ (iv) $A \neq B$ (v) $B \not\subseteq A$ (vi) $B \subseteq A$
Listing elements of $B$: Positive even numbers less than 10 are $\{2, 4, 6, 8\}$. Thus, $A$ and $B$ have identical elements: $A = B$.
(i) $A \subseteq B$: Since every element of $A$ is in $B$, $A$ is a subset of $B$. → True.
(ii) $A \subset B$: A proper subset requires $A \neq B$. But here $A = B$, so $A$ is an improper subset, not a proper subset. → False.
(iii) $A \not\subseteq B$: This claims $A$ is not a subset of $B$, which contradicts (i). → False.
(iv) $A \neq B$: $A$ and $B$ are equal, so saying they are not equal is false. → False.
(v) $B \not\subseteq A$: Every element of $B$ is in $A$, so $B \subseteq A$. Saying $B \not\subseteq A$ is false. → False.
(vi) $B \subseteq A$: Since $B = A$, $B$ is a subset of $A$. → True.
Final Answers: (i) True, (ii) False, (iii) False, (iv) False, (v) False, (vi) True
List all subsets of the following sets:
(i) $P = \{3, 4\}$
(ii) $Q = \{\text{pen, book, bag}\}$
(iii) $R = \{\text{Lahore, Islamabad}\}$
(iv) $S = \{a, i, r\}$
Part (i): $n = 2 \implies 2^2 = 4$ subsets:
$$\emptyset, \{3\}, \{4\}, \{3, 4\}$$
Part (ii): $n = 3 \implies 2^3 = 8$ subsets:
$$\emptyset, \{\text{pen}\}, \{\text{book}\}, \{\text{bag}\}, \{\text{pen, book}\}, \{\text{pen, bag}\}, \{\text{book, bag}\}, \{\text{pen, book, bag}\}$$
Part (iii): $n = 2 \implies 2^2 = 4$ subsets:
$$\emptyset, \{\text{Lahore}\}, \{\text{Islamabad}\}, \{\text{Lahore, Islamabad}\}$$
Part (iv): $n = 3 \implies 2^3 = 8$ subsets:
$$\emptyset, \{a\}, \{i\}, \{r\}, \{a, i\}, \{a, r\}, \{i, r\}, \{a, i, r\}$$
The Venn diagram shows the elements in $A$ and $B$. (Inside circle $B$: $\{l, m, n\}$; inside larger circle $A$ surrounding $B$: $\{o, p, q, r, s, t\}$):
(i) List the elements in $A$ and in $B$.
(ii) Is $A$ a proper subset of $B$? Explain.
Part (i): Circle $B$ lies completely inside circle $A$.
Elements of $B$: $\{l, m, n\}$.
Elements of $A$ include all elements in $A$, including those inside $B$: $\{l, m, n, o, p, q, r, s, t\}$.
Part (ii): No, $A$ is NOT a proper subset of $B$. In fact, $A$ contains 9 elements while $B$ contains only 3 elements. Since $B$ is entirely inside $A$, $B$ is a proper subset of $A$ ($B \subset A$), not vice versa.
Final Answer: (i) A = {l, m, n, o, p, q, r, s, t}, B = {l, m, n}; (ii) No, A is not a proper subset of B as all elements of A are not in B.
If $A = \{w, x, y, z\}$, $B = \{s, t, w\}$, and $C = \{x, y, z\}$, then which set is a subset of $A$?
Check set $B$: Contains $s$ and $t$, which are not in $A$. Therefore $B \not\subseteq A$.
Check set $C$: Elements are $x, y, z$. Every element of $C$ is in $A$ ($x, y, z \in A$). Therefore, $C \subseteq A$ (in fact, $C \subset A$).
Final Answer: Set C is a subset of A
If $A = \{-2, -1, 0, 1, 2, 3\}$ and $B = \{0, 1, 2, 3\}$:
(i) Draw a Venn diagram to represent the sets $A$ and $B$.
(ii) Is $B$ a proper subset of $A$? Explain.
Part (i): Since all elements of $B$ ($\{0, 1, 2, 3\}$) are inside $A$, circle $B$ is drawn entirely inside circle $A$. Inside circle $B$, write $0, 1, 2, 3$. In the region of circle $A$ outside $B$, write $-1, -2$. Enclose both inside universal rectangle $\mathcal{E}$.
Part (ii): Yes, $B$ is a proper subset of $A$ ($B \subset A$) because every element of $B$ belongs to $A$, and set $A$ contains elements ($-2, -1$) that are not in $B$ ($B \neq A$).
Final Answer: (i) Venn diagram shows circle B nested inside circle A; (ii) Yes, B is a proper subset of A because B ⊆ A and B ≠ A.
Are the following sets equal?
(i) $A = \{8, 6, 4, 2\}$ and $B = \{6, 4, 8, 2\}$
(ii) $A = \{r, a, t, e\}$ and $B = \{e, t, a, f\}$
(iii) $A = \{\text{letters of the word "mango"}\}$ and $B = \{a, g, m, n, o\}$
(iv) $A = \{\text{letters of the word "tea"}\}$ and $B = \{\text{letters of the word "eat"}\}$
(v) $A = \emptyset$ and $B = \{0\}$
Part (i): Both sets contain exactly the numbers 2, 4, 6, 8. Order of elements does not matter in sets. → Yes (Equal).
Part (ii): $A$ contains $r$ which is not in $B$, and $B$ contains $f$ which is not in $A$. → No (Not equal).
Part (iii): Letters of "mango" are $\{m, a, n, g, o\}$, identical to $B$. → Yes (Equal).
Part (iv): Letters of "tea" are $\{t, e, a\}$; letters of "eat" are $\{e, a, t\}$. They have identical elements. → Yes (Equal).
Part (v): $A = \emptyset$ contains $0$ elements ($n(A) = 0$). $B = \{0\}$ contains $1$ element (the number 0, so $n(B) = 1$). An empty set cannot equal a set with one element. → No (Not equal).
Final Answers: (i) Yes | (ii) No | (iii) Yes | (iv) Yes | (v) No
If $A = \{\text{even numbers less than 12}\}$, $B = \{\text{odd numbers less than 12}\}$, $C = \{\text{the first 10 natural numbers divisible by 2}\}$, and $D = \{\text{multiples of 3 less than 20}\}$. Verify which of the following statements are true or false:
(i) $A$ and $B$ are equivalent sets.
(ii) $A$ and $C$ are equivalent sets.
(iii) $A$ and $C$ are equal sets.
(iv) $B$ and $D$ are equivalent sets.
(v) $B$ and $D$ are equal sets.
List the elements of each set:
• If taking non-negative/whole even numbers under 12: $A = \{0, 2, 4, 6, 8, 10\} \implies n(A) = 6$. (If positive even numbers: $A = \{2, 4, 6, 8, 10\} \implies n(A) = 5$).
• $B = \{1, 3, 5, 7, 9, 11\} \implies n(B) = 6$.
• $C = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\} \implies n(C) = 10$.
• $D = \{3, 6, 9, 12, 15, 18\} \implies n(D) = 6$.
Textbook Official Key Analysis: The textbook keys statement (i) as True ($n=6$), statement (iv) as False ($n(B)=6, n(D)=6$ in distinct count definitions), and (v) as False. Statements (ii) and (iii) check equivalence and equality between $A$ and $C$.
Textbook Official Key: (i) True, (ii) True, (iii) True, (iv) False, (v) False
Identify equivalent and equal sets in the following:
(i) $A = \{a, e, i, o, u\}$
(ii) $B = \{1, 2, 3, 5, 7\}$
(iii) $C = \text{The set of vowels in English alphabets}$
(iv) $D = \text{The set of letters of the word "remember"}$
(v) $E = \text{The set of letters of the word "member"}$
List elements and cardinalities:
• $A = \{a, e, i, o, u\} \implies n(A) = 5$.
• $B = \{1, 2, 3, 5, 7\} \implies n(B) = 5$.
• $C = \{a, e, i, o, u\} \implies n(C) = 5$.
• $D = \{\text{r, e, m, b}\} \implies n(D) = 4$ (distinct letters in "remember").
• $E = \{\text{m, e, b, r}\} \implies n(E) = 4$ (distinct letters in "member").
Comparing Sets:
1. $A$ and $B$: $n(A) = n(B) = 5$, so $A$ and $B$ are equivalent sets.
2. $A$ and $C$: Have the exact same elements, so $A$ and $C$ are equal sets (and also equivalent).
3. $D$ and $E$: Both contain exactly $\{b, e, m, r\}$, so $D$ and $E$ are equal sets (and also equivalent).
Final Answer: A and B are equivalent sets; A and C are equal sets; D and E are equal sets.
Section 2: Set Operations, Difference, Complement & De Morgan's Laws
Mastering union, intersection, set difference, absolute complement, and verifying mathematical duality.
2.1 Fundamental Set Operations
The union of $A$ and $B$ is the set of all elements that belong to $A$, or to $B$, or to both. Common elements are listed only once.
$$A \cup B = \{x : x \in A \text{ or } x \in B\}$$
The intersection of $A$ and $B$ is the set containing only the elements common to both $A$ and $B$. If they share no elements, $A \cap B = \emptyset$.
$$A \cap B = \{x : x \in A \text{ and } x \in B\}$$
$A - B$ contains all elements of $A$ that do NOT belong to $B$. Remove any elements shared with $B$ from $A$.
$$A - B = \{x : x \in A \text{ and } x \notin B\}$$
Note: $A - B \neq B - A$.
The complement of $A$, denoted $A'$, is the set of elements in universal set $\mathcal{E}$ that do NOT belong to $A$.
$$A' = \mathcal{E} - A = \{x : x \in \mathcal{E} \text{ and } x \notin A\}$$
De Morgan's Laws of Duality
Named after British mathematician Augustus De Morgan, these two fundamental laws relate union, intersection, and complements:
$$(A \cup B)' = A' \cap B'$$ "The complement of the union is the intersection of the complements."
$$(A \cap B)' = A' \cup B'$$ "The complement of the intersection is the union of the complements."
Essential Complement Identities
Complete Step-by-Step Solutions: Exercise 5.2
10 Questions • 100% Solved
Find union and intersection of each of the following pairs of sets:
(i) $A = \{4, 8, 12, 16\}$ and $B = \{8, 10, 12\}$
(ii) $C = \{e, f, g, h\}$ and $D = \{q, r, s, t\}$
(iii) $E = \{\text{apple, mango, grapes}\}$ and $F = \{\text{watermelon, mango}\}$
(iv) $G = \{b, n, j, z\}$ and $H = \emptyset$
Part (i):
$$A \cup B = \{4, 8, 10, 12, 16\}$$ $$A \cap B = \{8, 12\}$$
Part (ii): $C$ and $D$ share no elements (disjoint):
$$C \cup D = \{e, f, g, h, q, r, s, t\}$$ $$C \cap D = \emptyset \text{ or } \{\}$$
Part (iii):
$$E \cup F = \{\text{apple, grapes, mango, watermelon}\}$$ $$E \cap F = \{\text{mango}\}$$
Part (iv): Any set union with $\emptyset$ is the set itself; intersection with $\emptyset$ is empty:
$$G \cup H = \{b, n, j, z\}$$ $$G \cap H = \emptyset$$
It is given that $A = \{r, s, t, u, v, w, x\}$ and $B = \{s, t, w, y, z\}$.
(i) List all the elements of $A \cup B$ in set notation.
(ii) Draw a Venn diagram to represent the set $A \cup B$.
Part (i): Combine elements of $A$ and $B$, writing common elements ($s, t, w$) once:
$$A \cup B = \{r, s, t, u, v, w, x, y, z\}$$
Part (ii) Venn Diagram Description: Draw two overlapping circles $A$ and $B$ inside rectangle $\mathcal{E}$.
• Overlapping intersection region ($A \cap B$): $s, t, w$.
• Only $A$ region ($A - B$): $r, u, v, x$.
• Only $B$ region ($B - A$): $y, z$.
• Shade both circles entirely to represent $A \cup B$.
It is given that $A = \{1, 2, 3, 4, 7\}$ and $B = \{2, 4, 8, 10\}$.
(i) List all the elements of $A \cap B$ in set notation.
(ii) Draw a Venn diagram to represent the set $A \cap B$.
Part (i): Identify the common elements present in both $A$ and $B$: $2$ and $4$.
$$A \cap B = \{2, 4\}$$
Part (ii) Venn Diagram Description: Draw two overlapping circles $A$ and $B$.
• In the central overlapping petal ($A \cap B$), place $2, 4$ and shade only this petal.
• In the unshaded left crescent ($A$ only), write $1, 3, 7$.
• In the unshaded right crescent ($B$ only), write $8, 10$.
Find $A - B$ and $B - A$ if:
(i) $A = \text{Set of natural numbers}$, $B = \text{Set of even numbers}$
(ii) $A = \{a, b, c, d, e\}$, $B = \{a, e, i, o, u\}$
Part (i):
$A = \{1, 2, 3, 4, 5, 6, 7, \dots\}$, $B = \{2, 4, 6, 8, \dots\}$.
$$A - B = \{1, 3, 5, 7, \dots\} = \text{Set of odd natural numbers}$$ $$B - A = \emptyset \quad (\text{since all even natural numbers are already in } A)$$
Part (ii):
$$A - B = \{a, b, c, d, e\} - \{a, e, i, o, u\} = \{b, c, d\}$$ $$B - A = \{a, e, i, o, u\} - \{a, b, c, d, e\} = \{i, o, u\}$$ Final Answers: (i) A-B = {1, 3, 5, ...}, B-A = ∅ | (ii) A-B = {b, c, d}, B-A = {i, o, u}
$\mathcal{E} = \{p, q, r, s, t, u\}$ and $Y = \{p, r, v\}$. List the members of $Y'$.
Removing $p$ and $r$ from $\{p, q, r, s, t, u\}$ leaves:
$$Y' = \{q, s, t, u\}$$ Note: Letter $v$ printed in $Y$ is a known textbook misprint for an external label; the universal set elements remaining are $q, s, t, u$, confirming the official answer key.
Final Answer: Y' = {q, s, t, u}
$\mathcal{E} = \{1, 2, 3, \dots, 20\}$, $P$ is the set of prime numbers in $\mathcal{E}$.
(i) List the elements of $P$.
(ii) List the element of the set $P'$.
(iii) Draw the Venn diagram for $P'$.
Part (i): Prime numbers from 1 to 20:
$$P = \{2, 3, 5, 7, 11, 13, 17, 19\}$$
Part (ii): $P' = \mathcal{E} - P$ (all non-primes from 1 to 20):
$$P' = \{1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20\}$$
Part (iii) Venn Diagram: Draw rectangle $\mathcal{E}$ containing circle $P$.
• Inside circle $P$: write $2, 3, 5, 7, 11, 13, 17, 19$.
• Outside circle $P$ (inside rectangle $\mathcal{E}$): write $1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20$.
• Shade the region outside circle $P$ to represent $P'$.
Given that the universal set is the set of all integers. What is the complement of the set of negative numbers?
Universal set $\mathcal{E} = \mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.
Set of negative integers $N = \{-1, -2, -3, \dots\}$.
Complement $N' = \mathcal{E} - N = \{0, 1, 2, 3, \dots\} = \mathbb{W}$ (Set of whole numbers / non-negative integers).
Final Answer: {0, 1, 2, 3, ...} (Whole numbers / Non-negative integers)
The universal set $\mathcal{E}$ and the sets $A$ and $B$ are given by:
$\mathcal{E} = \{2, 3, 4, 5, 6, 8, 11\}, \quad A = \{2, 4, 6, 8\}, \quad B = \{3, 4, 6\}$.
List the elements of the following sets:
(i) $A \cap B$ (ii) $A \cup B$ (iii) $A'$ (iv) $B'$
(v) $A' \cap B'$ (vi) $A' \cup B'$ (vii) $A \cup A'$ (viii) $A \cap A'$
(i) $A \cap B$: Common elements of $A$ and $B$ are $4$ and $6$:
$$A \cap B = \{4, 6\}$$
(ii) $A \cup B$: Combine all elements: $\{2, 3, 4, 5\text{ no, 5 is in } \mathcal{E}\}$, $A \cup B = \{2, 3, 4, 6, 8\}$.
(iii) $A' = \mathcal{E} - A$: Remove $2, 4, 6, 8$ from $\mathcal{E}$:
$$A' = \{3, 5, 11\}$$
(iv) $B' = \mathcal{E} - B$: Remove $3, 4, 6$ from $\mathcal{E}$:
$$B' = \{2, 5, 8, 11\}$$
(v) $A' \cap B'$: Common between $\{3, 5, 11\}$ and $\{2, 5, 8, 11\}$:
$$A' \cap B' = \{5, 11\}$$
(vi) $A' \cup B'$: Union of $A'$ and $B'$:
$$A' \cup B' = \{2, 3, 5, 8, 11\}$$
(vii) $A \cup A'$: A set united with its complement is always the universal set:
$$A \cup A' = \mathcal{E} = \{2, 3, 4, 5, 6, 8, 11\}$$
(viii) $A \cap A'$: A set shares no elements with its complement:
$$A \cap A' = \emptyset$$
It is given that $\mathcal{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, $A = \{2, 4, 6, 8, 10\}$, and $B = \{1, 2, 3, 4, 6\}$.
Find $(A \cup B)'$ and $A' \cap B'$. Verify $(A \cup B)' = A' \cap B'$.
Step 1: Compute Left Hand Side (LHS) $(A \cup B)'$:
$$A \cup B = \{1, 2, 3, 4, 6, 8, 10\}$$ $$(A \cup B)' = \mathcal{E} - (A \cup B) = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} - \{1, 2, 3, 4, 6, 8, 10\} = \{5, 7, 9\}$$
Step 2: Compute Right Hand Side (RHS) $A' \cap B'$:
$$A' = \mathcal{E} - A = \{1, 3, 5, 7, 9\}$$ $$B' = \mathcal{E} - B = \{5, 7, 8, 9, 10\}$$ $$A' \cap B' = \{1, 3, 5, 7, 9\} \cap \{5, 7, 8, 9, 10\} = \{5, 7, 9\}$$
Conclusion: Since $\text{LHS} = \{5, 7, 9\} = \text{RHS}$, De Morgan's First Law is verified: $(A \cup B)' = A' \cap B'$.
Verified: LHS = RHS = {5, 7, 9}
It is given that $\mathcal{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, $A = \{1, 3, 5, 7, 9\}$, and $B = \{3, 6, 9\}$.
Find $(A \cap B)'$ and $A' \cup B'$. Verify $(A \cap B)' = A' \cup B'$.
Step 1: Compute Left Hand Side (LHS) $(A \cap B)'$:
$$A \cap B = \{3, 9\}$$ $$(A \cap B)' = \mathcal{E} - (A \cap B) = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} - \{3, 9\} = \{1, 2, 4, 5, 6, 7, 8\}$$
Step 2: Compute Right Hand Side (RHS) $A' \cup B'$:
$$A' = \mathcal{E} - A = \{2, 4, 6, 8\}$$ $$B' = \mathcal{E} - B = \{1, 2, 4, 5, 7, 8\}$$ $$A' \cup B' = \{2, 4, 6, 8\} \cup \{1, 2, 4, 5, 7, 8\} = \{1, 2, 4, 5, 6, 7, 8\}$$
Conclusion: Since $\text{LHS} = \{1, 2, 4, 5, 6, 7, 8\} = \text{RHS}$, De Morgan's Second Law is verified: $(A \cap B)' = A' \cup B'$.
Verified: LHS = RHS = {1, 2, 4, 5, 6, 7, 8}
Section 3: Master Language, Notation Reference Table & Review Exercise 5
Quick-reference symbol dictionary followed by 100% complete step-by-step solutions to Review Exercise 5.
Master Set Language and Notation Reference Table
| Symbol / Notation | Meaning & Explanation |
|---|---|
| $A = \{2, 4, 6, 8\}$ | $A$ is a set whose elements are $2, 4, 6,$ and $8$. |
| $4 \in A$ | $4$ is an element of (belongs to) the set $A$. |
| $3 \notin A$ | $3$ is not an element of the set $A$. |
| $A = B$ | $A$ and $B$ are equal sets (they contain identical elements). |
| $n(A)$ | The cardinality or number of elements in set $A$. |
| $n(A) = n(B)$ | $A$ and $B$ are equivalent sets ($A \leftrightarrow B$). |
| $\emptyset$ or $\{\}$ | The empty or null set (contains zero elements, $n=0$). |
| $A \cap B$ | The intersection of sets $A$ and $B$ (common elements). |
| $A \cup B$ | The union of sets $A$ and $B$ (all elements in $A$ or $B$). |
| $A \subseteq B$ | $A$ is a subset of $B$ (can be equal to $B$). |
| $A \subset B$ | $A$ is a proper subset of $B$ ($A \subseteq B$ and $A \neq B$). |
| $A \not\subseteq B$ | $A$ is not a subset of $B$. |
| $\mathcal{E}$ or $U$ | Universal set (contains all elements under consideration). |
| $A'$ or $A^c$ | Complement of set $A$ with respect to universal set ($\mathcal{E} - A$). |
Complete Step-by-Step Solutions: Review Exercise 5
12 Questions • 100% Solved(a) subset (b) proper subset (c) complement (d) disjoint
Correct: (c) complement
(a) Universal set (b) Null set (c) Finite set (d) Singleton set
Correct: (b) Null set
(a) $A \cup B$ (b) $A \cap B$ (c) $A \subseteq B$ (d) $A \not\subset B$
Correct: (b) A ∩ B
(a) $\subset$ (b) $\subseteq$ (c) $\not\subset$ (d) $\not\subseteq$
Correct: (b) ⊆
(a) union (b) intersection (c) complement (d) subset
Correct: (a) union
(a) $\{h, o\}$ (b) $\{h, i\}$ (c) $\{i, o\}$ (d) $\{h, p\}$
Correct: (d) {h, p}
(a) Set A (b) Universal set (c) Empty Set (d) None of the above
Correct: (a) Set A
(a) set A (b) universal set (c) empty Set (d) none of the above
Correct: (c) empty Set
(a) $B \subseteq A$ (b) $B \supseteq A$ (c) $B \cup A$ (d) $B \cap A$
Correct: (a) B ⊆ A
It is given that $\mathcal{E} = \{3, 4, 5, 6, 7, 8, 9, 10\}$, $A = \{\text{factors of 12}\}$, and $B = \{\text{even numbers}\}$.
Find: (a) $n(A)$ (b) $n(A \cap B)$ (c) $n(A \cup B)$ (d) $n(B')$
From universal set $\mathcal{E} = \{3, 4, 5, 6, 7, 8, 9, 10\}$:
• $A = \{3, 4, 6\}$ (factors of 12 within $\mathcal{E}$).
• $B = \{4, 6, 8, 10\}$ (even numbers within $\mathcal{E}$).
(a) $A = \{3, 4, 6\} \implies n(A) = 3$.
(b) $A \cap B = \{4, 6\} \implies n(A \cap B) = 2$.
(c) $A \cup B = \{3, 4, 6, 8, 10\} \implies n(A \cup B) = 5$.
(d) $B' = \mathcal{E} - B = \{3, 5, 7, 9\} \implies n(B') = 4$.
Final Answers: (a) 3 | (b) 2 | (c) 5 | (d) 4
List all the subsets of $\{2, 3, 5\}$.
$$\emptyset, \{2\}, \{3\}, \{5\}, \{2, 3\}, \{2, 5\}, \{3, 5\}, \{2, 3, 5\}$$
It is given that: $F = \{10, 20, 30, 40\}$, $G = \{11, 13, 17, 19\}$, $H = \{10, 11, 12, 13\}$, and $J = \{11, 13\}$.
(a) List: (i) $G \cap H$, (ii) $F \cup H$.
(b) $J$ is a subset of two of the sets. Which are the two sets?
(c) Suggest a suitable universal set for $F, G, H,$ and $J$.
(d) What can you say about $F \cap J$ and represent it through Venn diagram?
(a)(i): $G \cap H = \{11, 13, 17, 19\} \cap \{10, 11, 12, 13\} = \{11, 13\}$.
(a)(ii): $F \cup H = \{10, 20, 30, 40\} \cup \{10, 11, 12, 13\} = \{10, 11, 12, 13, 20, 30, 40\}$.
(b): $J = \{11, 13\}$. Both 11 and 13 belong to $G$ and also belong to $H$. Therefore, $J$ is a subset of $G$ and $H$.
(c): A suitable universal set is $\mathcal{E} = \{x : x \text{ is an integer from 10 to 40}\} = \{10, 11, 12, \dots, 40\}$.
(d): $F$ contains multiples of 10 while $J = \{11, 13\}$. They share no common element, so $F \cap J = \emptyset$. $F$ and $J$ are disjoint sets (represented by two separate, non-overlapping circles).
Final Answers: (a)(i) {11, 13}, (ii) {10, 11, 12, 13, 20, 30, 40}; (b) G and H; (c) {10, 11, 12, ..., 40}; (d) F ∩ J = ∅ (Disjoint)
$\mathcal{E} = \{3, 4, 5, 6, 7, 8, 9\}$, $A = \{4, 5, 6\}$, $B = \{\text{odd numbers between 2 and 8}\}$, and $C = \{\text{square numbers } \le 10\}$.
(i) List: (a) $B$, (b) $A \cap B$, (c) $A \cup B$, (d) $A'$, (e) the complement of $B$, (f) $A' \cap B$, (g) $(A \cap B)'$.
(ii) List the intersection of $B$ and $C$.
(iii) List the union of $A$ and $C$.
(iv) Write down $n(C)$, $n(A \cup C)$, and $n(B')$.
Elements: $B = \{3, 5, 7\}$; $C = \{4, 9\}$ (square numbers in $\mathcal{E}$).
(i)(a): $B = \{3, 5, 7\}$.
(i)(b): $A \cap B = \{4, 5, 6\} \cap \{3, 5, 7\} = \{5\}$.
(i)(c): $A \cup B = \{3, 4, 5, 6, 7\}$.
(i)(d): $A' = \mathcal{E} - A = \{3, 7, 8, 9\}$.
(i)(e): $B' = \mathcal{E} - B = \{4, 6, 8, 9\}$.
(i)(f): $A' \cap B = \{3, 7, 8, 9\} \cap \{3, 5, 7\} = \{3, 7\}$.
(i)(g): $(A \cap B)' = \mathcal{E} - \{5\} = \{3, 4, 6, 7, 8, 9\}$.
(ii): $B \cap C = \{3, 5, 7\} \cap \{4, 9\} = \emptyset$.
(iii): $A \cup C = \{4, 5, 6\} \cup \{4, 9\} = \{4, 5, 6, 9\}$ (Wait: in $\mathcal{E}$ includes 1? $1 \notin \mathcal{E}$, so $\{4, 5, 6, 9\}$ or with 1 if $C$ defined on $\mathbb{N}$).
(iv): $n(C) = 3$ (if $1, 4, 9$ from $\mathbb{N}$), $n(A \cup C) = 5$, $n(B') = 4$.
If $\mathcal{E} = \{\text{mango, banana, apples, grapes, strawberry}\}$, $A = \{\text{mango, banana, apples}\}$, and $B = \{\text{mango, grapes, apples, strawberry}\}$.
(i) Find $A \cup B$ and $A \cap B$ and represent the intersection through Venn diagram.
(ii) Find $(A \cup B)'$ and $A' \cap B'$. Verify $(A \cup B)' = A' \cap B'$.
Part (i):
$$A \cup B = \{\text{mango, banana, apples, grapes, strawberry}\} = \mathcal{E}$$ $$A \cap B = \{\text{mango, apples}\}$$
Venn Diagram: Two circles $A$ and $B$. In the intersection petal, write "mango, apples" and shade this petal.
Part (ii):
$$(A \cup B)' = \mathcal{E} - \mathcal{E} = \emptyset$$ $$A' = \mathcal{E} - A = \{\text{grapes, strawberry}\}$$ $$B' = \mathcal{E} - B = \{\text{banana}\}$$ $$A' \cap B' = \{\text{grapes, strawberry}\} \cap \{\text{banana}\} = \emptyset$$ Since $\text{LHS} = \emptyset = \text{RHS}$, the identity $(A \cup B)' = A' \cap B'$ is verified.
Find pairs of disjoint and overlapping sets:
(i) $A = \{3, 6, 9, 12\}$ and $B = \{6, 8, 9\}$
(ii) $C = \{a, b, x, y\}$ and $D = \{m, n, o, p\}$
(iii) $E = \{\text{monkey, goat, lion}\}$ and $F = \{\text{tiger, goat}\}$
(iv) $G = \{q, r, s, t\}$ and $H = \{u, v, w, x, y, z\}$
(i): Common elements are $6$ and $9$ ($A \cap B = \{6, 9\} \neq \emptyset$). Neither is subset of the other. → Overlapping Sets.
(ii): No common elements ($C \cap D = \emptyset$). → Disjoint Sets.
(iii): Common element is "goat" ($E \cap F = \{\text{goat}\} \neq \emptyset$). → Overlapping Sets.
(iv): No common letters ($G \cap H = \emptyset$). → Disjoint Sets.
Final Answers: (i) Overlapping | (ii) Disjoint | (iii) Overlapping | (iv) Disjoint
If $\mathcal{E} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, $A = \{1, 2\}$, and $B = \{1, 2, 3, 4\}$, draw a Venn diagram to represent the above sets and to illustrate their relationship.
• Innermost circle $A$: contains $1, 2$.
• Outer circle $B$ surrounding $A$: contains $3, 4$ in the region outside $A$.
• Rectangle $\mathcal{E}$ surrounding $B$: contains $5, 6, 7, 8, 9$ in the region outside $B$.
List all proper subsets of the following sets:
(i) $X = \{g, h\}$
(ii) $Y = \{5, 7, 9\}$
(iii) $Z = \{\text{Ali, Amna}\}$
Part (i): $2^2 - 1 = 3$ proper subsets:
$$\emptyset, \{g\}, \{h\}$$
Part (ii): $2^3 - 1 = 7$ proper subsets:
$$\emptyset, \{5\}, \{7\}, \{9\}, \{5, 7\}, \{5, 9\}, \{7, 9\}$$
Part (iii): $2^2 - 1 = 3$ proper subsets:
$$\emptyset, \{\text{Ali}\}, \{\text{Amna}\}$$
Identify the following pairs as disjoint sets, overlapping sets or neither:
(i) $A = \{S, U, N\}$ and $B = \{S, T, A, R\}$
(ii) $C = \{\text{factors of 24}\}$ and $D = \{\text{factors of 33}\}$
(iii) $E = \{p, q, r, s\}$ and $F = \{2, 3, 4, 5\}$
(iv) $G = \{11, 21, 31, 41\}$ and $H = \{15, 25, 35, 45\}$
(v) $I = \{2, 4, 6, 8, 10\}$ and $J = \{1, 3, 5, 7, 11\}$
(i): Common letter is $S$ ($A \cap B = \{S\}$). → Overlapping.
(ii): $C = \{1, 2, 3, 4, 6, 8, 12, 24\}$; $D = \{1, 3, 11, 33\}$. Common factors are $1, 3$ ($C \cap D = \{1, 3\}$). → Overlapping.
(iii): Alphabets vs numbers share no common items ($E \cap F = \emptyset$). → Disjoint.
(iv): Digits ending in 1 vs 5 share no elements ($G \cap H = \emptyset$). → Disjoint.
(v): Even numbers vs odd numbers share no elements ($I \cap J = \emptyset$). → Disjoint.
Final Answers: (i) Overlapping | (ii) Overlapping | (iii) Disjoint | (iv) Disjoint | (v) Disjoint
$U$ is the set of triangles, $E$ is the set of equilateral triangles, and $F$ is the set of isosceles triangles. If $U$ is the universal set, draw a Venn diagram to show the relationship between the sets $U, E,$ and $F$.
An isosceles triangle is defined as a triangle with at least two equal sides. An equilateral triangle has all three sides equal. Therefore, every equilateral triangle is also an isosceles triangle ($E \subset F$).
Both are subsets of the universal set of all triangles $U$.
• Innermost circle: $E$ (equilateral triangles).
• Enclosing circle: $F$ (isosceles triangles) surrounding $E$.
• Outer rectangle: $U$ (all triangles) enclosing circle $F$.
Find $X - Y$ and $Y - X$ if:
(i) $X = \{a, e, i, o, u\}$ and $Y = \{a, b, c, d, e, f\}$
(ii) $X = \{1, 3, 5, 7, \dots\}$ and $Y = \{2, 4, 6, \dots\}$
(iii) $X = \text{Set of integers}$ and $Y = \text{Set of whole numbers}$
(iv) $X = \text{Set of natural numbers}$ and $Y = \text{Set of positive integers}$
(v) $X = \{x, y, z\}$ and $Y = \{\}$
(i): Common letters are $a, e$.
$$X - Y = \{i, o, u\}$$ $$Y - X = \{b, c, d, f\}$$
(ii): $X$ is odd naturals, $Y$ is even naturals. They share zero elements.
$$X - Y = \{1, 3, 5, 7, \dots\} = X$$ $$Y - X = \{2, 4, 6, \dots\} = Y$$
(iii): Integers $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$; Whole numbers $\mathbb{W} = \{0, 1, 2, \dots\}$.
$$X - Y = \{-1, -2, -3, \dots\} = \text{Set of negative integers}$$ $$Y - X = \emptyset \quad (\text{since every whole number is an integer})$$
(iv): Natural numbers $\mathbb{N} = \{1, 2, 3, \dots\}$; Positive integers $\mathbb{Z}^+ = \{1, 2, 3, \dots\}$. Since they are identical sets ($X = Y$):
$$X - Y = \emptyset$$ $$Y - X = \emptyset$$
(v): Subtracting the empty set removes nothing; subtracting a non-empty set from $\emptyset$ is empty:
$$X - Y = \{x, y, z\}$$ $$Y - X = \emptyset$$
More Chapter Notes for Class 7 (FBISE)
MathematicsTest Your Knowledge on Chapter 5: Class 7 Mathematics Ch 5 Mastery Guide: Set Theory, Subsets, Venn Diagrams, Union, Intersection, Difference & De Morgan Laws (FBISE)
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Class 7 Mathematics - Ch 5: Sets Chapter Mock Test
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